Henry Ernest Dudeney/Puzzles and Curious Problems/229 - Geometrical Progression/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $229$

Geometrical Progression

Write out a series of whole numbers in geometrical progression, starting from $1$,

so that the numbers should add up to a square.

Thus, $1 + 2 + 4 + 8 + 16 + 32 = 63$.

But this is one short of being a square.

Solution

$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4 = 11^2$

$400 = 7^0 + 7^1 + 7^2 + 7^3 = 20^2$

Proof

This is explored in Square Numbers which are Sum of Consecutive Powers.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $229$. -- Geometrical Progression
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $243$. Geometrical Progression